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Journal of Automated Reasoning

, Volume 59, Issue 4, pp 425–454 | Cite as

Formally Proving Size Optimality of Sorting Networks

  • Luís Cruz-Filipe
  • Kim S. Larsen
  • Peter Schneider-Kamp
Article

Abstract

Recent successes in formally verifying increasingly larger computer-generated proofs have relied extensively on (a) using oracles, to find answers for recurring subproblems efficiently, and (b) extracting formally verified checkers, to perform exhaustive case analysis in feasible time. In this work we present a formal verification of optimality of sorting networks on up to 9 inputs, making it one of the largest computer-generated proofs that has been formally verified. We show that an adequate pre-processing of the information provided by the oracle is essential for feasibility, as it improves the time required by our extracted checker by several orders of magnitude.

Keywords

Interactive theorem proving Large-scale proofs Program extraction Sorting networks 

Mathematics Subject Classification

68T15 68Q60 68N18 68Q25 68Q17 

Notes

Acknowledgements

The authors would like to thank Mike Codish for all the fruitful discussions on sorting networks and Pierre Letouzey for helping with the extraction mechanism and giving suggestions on how to improve the performance of the extracted code.

References

  1. 1.
    Appel, K., Haken, W.: Every planar map is four colorable. Part I: discharging. Ill. J. Math. 21, 429–490 (1977)zbMATHGoogle Scholar
  2. 2.
    Appel, K., Haken, W.: The four color proof suffices. Math. Intell. 8(1), 10–20 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Appel, K., Haken, W., Koch, J.: Every planar map is four colorable. Part II: reducibility. Ill. J. Math. 21, 491–567 (1977)zbMATHGoogle Scholar
  4. 4.
    Bertot, Y., Castéran, P.: Interactive Theorem Proving and Program Development, Texts in Theoretical Computer Science. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  5. 5.
    Blanqui, F., Koprowski, A.: CoLoR: a Coq library on well-founded rewrite relations and its application to the automated verification of termination certificates. Math. Struct. Comput. Sci. 21, 827–859 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.): Interactive Theorem Proving, ITP 2013, Proceedings, LNCS, vol. 7998. Springer, Berlin (2013)Google Scholar
  7. 7.
    Bundala, D., Závodný, J.: Optimal sorting networks. In: A.H. Dediu, C. Martín-Vide, J.L. Sierra-Rodríguez, B. Truthe (eds.) LATA, LNCS, vol. 8370, pp. 236–247. Springer, Berlin (2014)Google Scholar
  8. 8.
    Chung, M.J., Ravikumar, B.: Bounds on the size of test sets for sorting and related networks. Discrete Math. 81(1), 1–9 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Claret, G., González-Huesca, L., Régis-Gianas, Y., Ziliani, B.: Lightweight proof by reflection using a posteriori simulation of effectful computation. In: Blazy et al. [6], pp. 67–83Google Scholar
  10. 10.
    Codish, M., Cruz-Filipe, L., Frank, M., Schneider-Kamp, P.: Twenty-five comparators is optimal when sorting nine inputs (and twenty-nine for ten). In: ICTAI 2014, pp. 186–193. IEEE (2014)Google Scholar
  11. 11.
    Codish, M., Cruz-Filipe, L., Frank, M., Schneider-Kamp, P.: Sorting nine inputs requires twenty-five comparisons. J. Comput. Syst. Sci. 82(3), 551–563 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Contejean, E., Courtieu, P., Forest, J., Pons, O., Urbain, X.: Automated certified proofs with CiME3. In: Schmidt-Schauß, M. (ed.) RTA 2011, LIPIcs, vol. 10, pp. 21–30. Schloss Dagstuhl (2011). http://www.dagstuhl.de/dagpub/978-3-939897-30-9
  13. 13.
    Cruz-Filipe, L., Letouzey, P.: A large-scale experiment in executing extracted programs. Electron. Notes Comput. Sci. 151(1), 75–91 (2006)CrossRefzbMATHGoogle Scholar
  14. 14.
    Cruz-Filipe, L., Schneider-Kamp, P.: Formalizing size-optimal sorting networks: extracting a certified proof checker. In: C. Urban, X. Zhang (eds.) ITP 2015, LNCS, vol. 9236, pp. 154–169. Springer, Berlin (2015)Google Scholar
  15. 15.
    Cruz-Filipe, L., Schneider-Kamp, P.: Optimizing a certified proof checker for a large-scale computer-generated proof. In: Kerber, M., Carette, J., Kaliszyk, C., Rabe, F., Sorge, V. (eds.) CICM 2015, LNAI, vol. 9150, pp. 55–70. Springer (2015)Google Scholar
  16. 16.
    Cruz-Filipe, L., Wiedijk, F.: Hierarchical reflection. In: Slind, K., Bunker, A., Gopalakrishnan,G. (eds.) TPHOLs, LNCS, vol. 3223, pp. 66–81. Springer, Berlin (2004)Google Scholar
  17. 17.
    Darbari, A., Fischer, B., Marques-Silva, J.: Industrial-strength certified SAT solving through verified SAT proof checking. In: ICTAC, Lecture Notes in Computer Science, vol. 6255, pp. 260–274. Springer, Berlin (2010)Google Scholar
  18. 18.
    Erkök, L., Matthews, J.: Using Yices as an automated solver in Isabelle/HOL. In: Automated Formal Methods’08, pp. 3–13. ACM Press, Princeton, NJ, USA (2008)Google Scholar
  19. 19.
    Floyd, R., Knuth, D.: The Bose–Nelson sorting problem. In: Srivastava, J. N. (ed.) A Survey of Combinatorial Theory, pp. 163–172. North-Holland, Amsterdam (1973)Google Scholar
  20. 20.
    Fouilhé, A., Monniaux, D., Périn, M.: Efficient generation of correctness certificates for the abstract domain of polyhedra. In: Logozzo, F., ähndrich, M.F (eds.) SAS 2013, LNCS, vol. 7935, pp. 345–365. Springer, Berlin (2013)Google Scholar
  21. 21.
    Gonthier, G.: Formal proof—the four-color theorem. Not. AMS 55(11), 1382–1393 (2008)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Harrison, J.: HOL Light: a tutorial introduction. In: Srivas, M.K., Camilleri, A.J. (eds.) FMCAD, Lecture Notes in Computer Science, vol. 1166, pp. 265–269. Springer, Berlin (1996)Google Scholar
  23. 23.
    Harrison, J., Théry, L.: A skeptic’s approach to combining HOL and maple. J. Autom. Reason. 21(3), 279–294 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Heule, M., Kullmann, O., Marek, V.: Solving and verifying the boolean pythagorean triples problem via cube-and-conquer. In: Creignou, N., Le Berre, D. (eds.) SAT 2016, Lecture Notes in Computer Science, vol. 9710, pp. 228–245. Springer, Berlin (2016)Google Scholar
  25. 25.
    Johnson, D.S.: The np-completeness column: an ongoing guide. J. Algorithms 3, 288–300 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Knuth, D.: The Art of Computer Programming, Volume III: Sorting and Searching. Springer, Reading (1973)zbMATHGoogle Scholar
  27. 27.
    Konev, B., Lisitsa, A.: A SAT attack on the Erdős discrepancy conjecture. In: C. Sinz, U. Egly (eds.) SAT 2014, LNCS, vol. 8561, pp. 219–226. Springer, Berlin (2014)Google Scholar
  28. 28.
    Krebbers, R., Spitters, B.: Computer certified efficient exact reals in Coq. In: Davenport, J., Farmer, W., Urban, J., Rabe, F. (eds.) Calculemus 2011, LNCS, vol. 6824, pp. 90–106. Springer, Berlin (2011)Google Scholar
  29. 29.
    Leroy, X.: Formal verification of a realistic compiler. Commun. ACM 52(7), 107–115 (2009)CrossRefGoogle Scholar
  30. 30.
    Letouzey, P.: Extraction in Coq: an overview. In: Beckmann, A., Dimitracopoulos, C., Löwe, B. (eds.) CiE 2008, LNCS, vol. 5028, pp. 359–369. Springer, Berlin (2008)Google Scholar
  31. 31.
    McBride, C.: Elimination with a motive. In: Callaghan, P., Luo, Z., McKinna, J., Pollack, R. (eds.) TYPES, LNCS, vol. 2277, pp. 197–216. Springer, Berlin (2002)Google Scholar
  32. 32.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL—A Proof Assistant for Higher-Order Logic. Springer, Berlin (2002)zbMATHGoogle Scholar
  33. 33.
    Norell, U.: Towards a practical programming language based on dependent type theory. Ph.D. thesis, Department of Computer Science and Engineering, Chalmers University of Technology (2007)Google Scholar
  34. 34.
    O’Connor, R.: Certified exact transcendental real number computation in Coq. In: Mohamed, O., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008, LNCS, vol. 5170, pp. 246–261. Springer, Berlin (2008)Google Scholar
  35. 35.
    Oury, N.: Observational equivalence and program extraction in the Coq proof assistant. In: Hofmann, M. (ed.) TLCA 2003, LNCS, vol. 2701, pp. 271–285. Springer, Berlin (2003)Google Scholar
  36. 36.
    Parberry, I.: A computer-assisted optimal depth lower bound for nine-input sorting networks. Math. Syst. Theory 24(2), 101–116 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Sternagel, C., Thiemann, R.: The certification problem format. In: Benzmüller, C., Paleo, B. (eds.) UITP 2014, EPTCS, vol. 167, pp. 61–72 (2014)Google Scholar
  38. 38.
    Thiemann, R.: Formalizing bounded increase. In: Blazy et al. [6], pp. 245–260Google Scholar
  39. 39.
    van Emde Boas, P.: Preserving order in a forest in less than logarithmic time and linear space. Inf. Process. Lett. 6(3), 80–82 (1977)CrossRefzbMATHGoogle Scholar
  40. 40.
    Van Voorhis, D.: Toward a lower bound for sorting networks. In: Miller, R., Thatcher, J. (eds.) Complexity of Computer Computations, The IBM Research Symposia Series, pp. 119–129. Plenum Press, New York (1972)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdense MDenmark

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